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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
lemma finiteSAT_of_hall {n k m : ℕ} (w : Fin m → Clause n k)
(hh : ∀ I : Finset (Fin m), I.card ≤ (I.biUnion (fun j => (w j).1.1)).card) :
finiteSAT w m := by
obtain ⟨f,hf,hm⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective
(fun j => (w j).1.1)).mp hh
let a : Assignment n := Function.extend f (fun j => (w j).2 ⟨f j,hm j⟩) (fun _ => false)
refine ⟨a,fun j _ => ⟨⟨f j,hm j⟩,?_⟩⟩
exact hf.extend_apply _ _ j
lemma deficient_witness {n k m : ℕ} (w : Fin m → Clause n k) (hw : ¬finiteSAT w m) :
∃ W : Finset (Fin n), ∃ I : Finset (Fin m), I.card = W.card+1 ∧
∀ j ∈ I, (w j).1.1 ⊆ W := by
have hh : ¬∀ I : Finset (Fin m), I.card ≤ (I.biUnion (fun j => (w j).1.1)).card :=
fun h => hw (finiteSAT_of_hall w h)
push Not at hh
obtain ⟨J,hJ⟩ := hh
obtain ⟨I,hIJ,hI⟩ := Finset.exists_subset_card_eq (Nat.succ_le_of_lt hJ)
refine ⟨J.biUnion (fun j => (w j).1.1),I,hI,fun j hj v hv => ?_⟩
exact Finset.mem_biUnion.mpr ⟨j,hIJ hj,hv⟩
lemma favg_support {n k : ℕ} (f : {s : Finset (Fin n) // s.card = k} → ℝ) :
favg (fun c : Clause n k => f c.1) = favg f := by
unfold favg
change (∑ c : (s : {s : Finset (Fin n) // s.card = k}) × (s.1 → Bool), f c.1) /
Fintype.card (Clause n k) = _
rw [Fintype.sum_sigma, clause_card, support_card]
have he (s : {s : Finset (Fin n) // s.card = k}) : (∑ _ : s.1 → Bool, f s) = 2^k*f s := by
simp [s.property]
simp only [he, ← Finset.mul_sum, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
have hp : (2:ℝ)^k ≠ 0 := by positivity
field_simp
lemma containing_support_probability {n k : ℕ} (W : Finset (Fin n)) :
favg (fun c : Clause n k => if c.1.1 ⊆ W then (1:ℝ) else 0) =
(W.card.choose k : ℝ)/(n.choose k) := by
have he0 := favg_support (fun s : {s : Finset (Fin n) // s.card = k} =>
if s.val ⊆ W then (1:ℝ) else 0)
rw [he0]
unfold favg
rw [support_card, Finset.sum_boole]
congr 1
have he : {s : {s : Finset (Fin n) // s.card = k} // s.1 ⊆ W} ≃
{s : Finset (Fin n) // s ⊆ W ∧ s.card = k} :=
{ toFun := fun s => ⟨s.1.1,s.2,s.1.2⟩
invFun := fun s => ⟨⟨s.1,s.2.2⟩,s.2.1⟩
left_inv := fun _ => rfl
right_inv := fun _ => rfl }
have hc := Fintype.card_congr he
rw [Fintype.card_of_subtype (W.powersetCard k) (by intro s; simp)] at hc
simpa only [Fintype.card_subtype, Finset.card_powersetCard] using congrArg (fun x : ℕ => (x:ℝ)) hc
lemma containing_support_bound {n k : ℕ} (hkn : k ≤ n) (W : Finset (Fin n)) :
favg (fun c : Clause n k => if c.1.1 ⊆ W then (1:ℝ) else 0) ≤
((W.card : ℝ)/n)^k := by
rw [containing_support_probability]
have hb : W.card ≤ n := (Finset.card_le_univ W).trans_eq (Fintype.card_fin n)
have hh := mul_le_mul_of_nonneg_left (killRatio_le_density_pow hb hkn)
(show (0:ℝ) ≤ 2^k by positivity)
have he1 : (2:ℝ)^k*killRatio n W.card k = (W.card.choose k : ℝ)/(n.choose k) := by
unfold killRatio
field_simp
have he2 : (2:ℝ)^k*((W.card:ℝ)/(2*n))^k = ((W.card:ℝ)/n)^k := by
rw [← mul_pow]
congr 1
field_simp
rw [he1,he2] at hh
exact hh
lemma favg_pi_on.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [Fintype ι] [DecidableEq ι] [Fintype α] [Nonempty α]
(I : Finset ι) (f : α → ℝ) :
favg (fun w : ι → α => ∏ i ∈ I, f (w i)) = (favg f)^I.card := by
have he (w : ι → α) : (∏ i ∈ I, f (w i)) = ∏ i, if i ∈ I then f (w i) else 1 := by
symm
rw [← Finset.prod_filter]
congr 1
ext i
simp
simp only [he]
change favg (fun w : ι → α => ∏ i, (fun i a => if i ∈ I then f a else 1) i (w i)) = _
rw [favg_pi_product (fun i a => if i ∈ I then f a else 1)]
have hh (i : ι) : favg (fun a => if i ∈ I then f a else 1) = if i ∈ I then favg f else 1 := by
split_ifs <;> simp [favg_const]
simp only [hh]
rw [← Finset.prod_filter]
have he' : Finset.univ.filter (fun i => i ∈ I) = I := by ext i; simp
rw [he', Finset.prod_const]
lemma hall_union_pointwise {n k m : ℕ} (w : Fin m → Clause n k) :
(if ¬finiteSAT w m then (1:ℝ) else 0) ≤
∑ W : Finset (Fin n), ∑ I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0 := by
have hn (W : Finset (Fin n)) (I : Finset (Fin m)) :
0 ≤ ∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0 := by positivity
by_cases hw : finiteSAT w m
· rw [ite_eq_right (not_not.mpr hw)]
exact Finset.sum_nonneg (fun W _ => Finset.sum_nonneg (fun I _ => hn W I))
· rw [ite_eq_left hw]
obtain ⟨W,I,hI,hwI⟩ := deficient_witness w hw
have hi : I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1) := by simp [hI]
calc
1 = ∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0 := by
symm
apply Finset.prod_eq_one
intro i hi
simp [hwI i hi]
_ ≤ ∑ J ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
∏ i ∈ J, if (w i).1.1 ⊆ W then (1:ℝ) else 0 :=
Finset.single_le_sum (fun J _ => hn W J) hi
_ ≤ _ := Finset.single_le_sum
(f := fun W : Finset (Fin n) => ∑ I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0)
(fun W _ => Finset.sum_nonneg (fun I _ => hn W I)) (Finset.mem_univ W)
lemma hall_term_bound {n m t : ℕ} (hn : 0 < n) (htn : t ≤ n) {a : ℝ}
(ha : 0 ≤ a) (hm : (m:ℝ) ≤ a*n) :
(n.choose t : ℝ)*(m.choose (t+1) : ℝ)*((t:ℝ)/n)^(2*(t+1)) ≤
a*(Real.exp 2*a)^t := by
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hn0 : (n:ℝ) ≠ 0 := ne_of_gt hn'
have ht : (0:ℝ) ≤ t := by positivity
have htn' : (t:ℝ)/n ≤ 1 := (div_le_one hn').mpr (by exact_mod_cast htn)
have hmn : (m:ℝ)/n ≤ a := (div_le_iff₀ hn').mpr hm
have hfac : (0:ℝ) < t.factorial := by positivity
have hfac' : (0:ℝ) < (t+1).factorial := by positivity
calc
_ ≤ ((n:ℝ)^t/t.factorial)*((m:ℝ)^(t+1)/(t+1).factorial)*((t:ℝ)/n)^(2*(t+1)) := by
gcongr
· exact Nat.choose_le_pow_div t n
· exact Nat.choose_le_pow_div (t+1) m
_ = ((m:ℝ)/n)^(t+1)*((t:ℝ)/n)*
((t:ℝ)^t/t.factorial)*((t:ℝ)^(t+1)/(t+1).factorial) := by
rw [show 2*(t+1) = t+t+2 by omega]
simp only [pow_add, pow_succ, div_pow]
field_simp
_ ≤ a^(t+1)*1*(Real.exp t)*(Real.exp t) := by
gcongr
· exact Real.pow_div_factorial_le_exp t ht t
· exact Real.pow_div_factorial_le_exp t ht (t+1)
_ = a*(Real.exp 2*a)^t := by
rw [mul_one, mul_assoc, ← Real.exp_add,
show (t:ℝ)+(t:ℝ) = (t:ℝ)*2 by ring, Real.exp_nat_mul, mul_pow, pow_succ]
ring
lemma finite_geometric_half {r : ℝ} (hr : 0 ≤ r) (hr1 : r ≤ 1/2) (N : ℕ) :
(∑ t ∈ Finset.range N, r^t) ≤ 2 := by
have hn : ‖r‖ < 1 := by rw [Real.norm_eq_abs, abs_of_nonneg hr]; linarith
calc
_ ≤ ∑' t : ℕ, r^t := (summable_geometric_of_norm_lt_one hn).sum_le_tsum _
(fun t _ => pow_nonneg hr t)
_ = (1-r)⁻¹ := tsum_geometric_of_norm_lt_one hn
_ ≤ 2 := by
rw [inv_eq_one_div]
apply (div_le_iff₀ (by linarith)).mpr
linarith
lemma hall_sum_bound {n m : ℕ} (hn : 0 < n) {a : ℝ} (ha : 0 ≤ a)
(ha' : Real.exp 2*a ≤ 1/2) (hm : (m:ℝ) ≤ a*n) :
(∑ t ∈ Finset.range (n+1), (n.choose t : ℝ)*(m.choose (t+1) : ℝ)*
((t:ℝ)/n)^(2*(t+1))) ≤ 2*a := by
calc
_ ≤ ∑ t ∈ Finset.range (n+1), a*(Real.exp 2*a)^t := by
apply Finset.sum_le_sum
intro t ht
exact hall_term_bound hn (by have := Finset.mem_range.mp ht; omega) ha hm
_ = a*(∑ t ∈ Finset.range (n+1), (Real.exp 2*a)^t) := by rw [Finset.mul_sum]
_ ≤ a*2 := mul_le_mul_of_nonneg_left (finite_geometric_half (by positivity) ha' _) ha
_ = _ := by ring
lemma hall_probability_bound {n k m : ℕ} (hk : 2 ≤ k) (hkn : k ≤ n)
{a : ℝ} (ha : 0 ≤ a) (ha' : Real.exp 2*a ≤ 1/2) (hm : (m:ℝ) ≤ a*n) :
favg (fun w : Fin m → Clause n k => if ¬finiteSAT w m then (1:ℝ) else 0) ≤ 2*a := by
let := clause_nonempty n k hkn
have hn : 0 < n := by omega
have hn' : (0:ℝ) < n := by exact_mod_cast hn
have hi (W : Finset (Fin n)) (I : Finset (Fin m))
(hI : I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1)) :
favg (fun w : Fin m → Clause n k => ∏ i ∈ I,
if (w i).1.1 ⊆ W then (1:ℝ) else 0) ≤ ((W.card:ℝ)/n)^(2*(W.card+1)) := by
rw [favg_pi_on I (fun c : Clause n k => if c.1.1 ⊆ W then (1:ℝ) else 0)]
have hic := (Finset.mem_powersetCard.mp hI).2
rw [hic, pow_mul]
apply pow_le_pow_left₀ (favg_nonneg (fun _ => by positivity))
apply (containing_support_bound hkn W).trans
have hr : (W.card:ℝ)/n ≤ 1 := (div_le_one hn').mpr (by exact_mod_cast (show W.card ≤ n by simpa using Finset.card_le_univ W))
exact pow_le_pow_of_le_one (by positivity) hr hk
calc
_ ≤ favg (fun w : Fin m → Clause n k =>
∑ W : Finset (Fin n), ∑ I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0) := favg_mono hall_union_pointwise
_ = ∑ W : Finset (Fin n), ∑ I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
favg (fun w : Fin m → Clause n k =>
∏ i ∈ I, if (w i).1.1 ⊆ W then (1:ℝ) else 0) := by
rw [favg_sum]
apply Finset.sum_congr rfl
intro W _
exact favg_sum _ _
_ ≤ ∑ W : Finset (Fin n),
(m.choose (W.card+1):ℝ)*((W.card:ℝ)/n)^(2*(W.card+1)) := by
apply Finset.sum_le_sum
intro W _
calc
_ ≤ ∑ I ∈ (Finset.univ : Finset (Fin m)).powersetCard (W.card+1),
((W.card:ℝ)/n)^(2*(W.card+1)) := Finset.sum_le_sum (hi W)
_ = _ := by simp
_ = ∑ t ∈ Finset.range (n+1), (n.choose t:ℝ)*(m.choose (t+1):ℝ)*
((t:ℝ)/n)^(2*(t+1)) := by
have he : (Finset.univ : Finset (Finset (Fin n))) =
(Finset.univ : Finset (Fin n)).powerset := by ext W; simp
rw [he,Finset.sum_powerset_apply_card (fun t : ℕ =>
(m.choose (t+1):ℝ)*((t:ℝ)/n)^(2*(t+1)))]
simp only [Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_assoc]
_ ≤ 2*a := hall_sum_bound hn ha ha' hm
end RandomKSAT
end
end OAI
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